Re: [PrimeNumbers] Re: Prime number "spans"
> > This notion is derived from the familiar prime gaps: The "span" of aIf we let span(x) to be defined for all natural numbers as the smaller of
> > prime is the smaller of the two gaps adjacent to it. For example,
> > span(839) is min(839-829-1,853-839-1) = min(9,13) = 9. One question
> > that arises immediately: can one easily prove the existence of
> > arbitrarily long spans (easy for gaps, using the factorial
> > construction)? With only a casual look, a method is not obvious.
> > Another question: are there any known tables of maximal spans?
> The factorial construction of proving a long gap is symmetric.
> If p is prime, then p# + 2, P# + 3, ...... up to p# + (p+1) are
> necessarily composite. and also, p# -2, p#-3, p#-4, .... down to
> p#-(p+1) are necessarily composite.
> So to prove arbitrarily long "spans", you need only prove that there are
> infinitely many composites of the form (p# +1) or (p# - 1).
the "gaps" between this number and the previous or next prime, it'd be
trivial to show that it can attain arbitrarily large values. It can be
done unconditionally using the factorial construction --- since the number
x=((2n+2)! + n + 2) has at least "n" composites on each side, span(x) must
be at least "n". For example, for n=3, composites 8!+2, 8!+3, 8!+4 precede
8!+5 and 8!+6, 8!+7, 8!+8 follow it.
The problem is to show that it can grow indefinitely even if we restrict
ourselves to prime values of "x".
[Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278